To graph the linear inequality $-2x - 4y \geq 8$, follow these steps:

Step 1: Rewrite the inequality in slope-intercept form

The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Let's first turn the given inequality into equality:

$-2x - 4y = 8$

Now, solve for $y$:

$-4y = 2x + 8$

Divide both sides by $-4$:

$y = -\frac{1}{2}x - 2$

Now we have the line equation $y = -\frac{1}{2}x - 2$.

Step 2: Graph the boundary line

Since the inequality includes "greater than or equal to" ($\geq$), the boundary line will be solid. This shows that points on the line are included in the solution set. The line has:

• Slope $m = -\frac{1}{2}$
• Y-intercept $b = -2$

To graph the line:

1. Plot the y-intercept (0, -2).
2. From that point, use the slope $-\frac{1}{2}$, which means "down 1 unit" and "right 2 units," to find additional points on the line.
3. Draw the solid line through these points.

Step 3: Shade the region

Since the inequality is $-2x - 4y \geq 8$, which translates to $y \geq -\frac{1}{2}x - 2$, you need to shade the region above the line because it includes values where $y$ is greater than or equal to the expression on the right-hand side.

Final Graph

1. Solid line for $y = -\frac{1}{2}x - 2$.
2. Shade the region above the line.

This is the graph for the inequality $-2x - 4y \geq 8$.

Would you like more details or need clarifications on any step?

Here are some follow-up questions:

1. What is the significance of the slope in a linear equation?
2. How does shading differ when using a "greater than" versus a "less than" inequality?
3. Can you explain how to determine whether to shade above or below a line?
4. How do you convert a standard form inequality into slope-intercept form?
5. What is the difference between solid and dashed boundary lines?

Tip: Always double-check if the inequality includes the boundary line (solid) or excludes it (dashed) to ensure correct shading.